If a tangent to the circle $${x^2} + {y^2} = 1$$   intersects the coordinate axes at distinct points $$P$$ and $$Q,$$  then the locus of the mid-point of $$PQ$$  is:

Solution :
Let any tangent to circle $${x^2} + {y^2} = 1$$   is $$x\,\cos \,\theta + y\,\sin \,\theta = 1$$
Since, $$P$$ and $$Q$$ are the point of intersection on the co-ordinate axes.
Then $$P \equiv \left( {\frac{1}{{\cos \,\theta }},\,0} \right)\,\,\& \,\,Q \equiv \left( {0,\,\frac{1}{{\sin \,\theta }}} \right)$$
$$\therefore $$ mid-point of $$PQ$$  be
$$\eqalign{ & M \equiv \left( {\frac{1}{{2\,\cos \,\theta }},\,\,\frac{1}{{2\,\sin \,\theta }}} \right) \equiv \left( {h,\,k} \right) \cr & \Rightarrow \cos \,\theta = \frac{1}{{2h}}.....(1) \cr & \Rightarrow \sin \,\theta = \frac{1}{{2k}}.....(2) \cr} $$
Now squaring and adding equation (1) and (2)
$$\eqalign{ & \frac{1}{{{h^2}}} + \frac{1}{{{k^2}}} = 4 \cr & \Rightarrow {h^2} + {k^2} = 4{h^2}{k^2} \cr} $$
$$\therefore $$ locus of $$M$$ is : $${x^2} + {y^2} - 4{x^2}{y^2} = 0$$